Axiom ax-mulass 8110

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 8068. Proofs should normally use mulass 8138 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 8005	. . . 4 class ℂ
3	1, 2	wcel 2200	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2200	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2200	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 1002	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 8012	. . . . 5 class ·
10	1, 4, 9	co 6007	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 6007	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 6007	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 6007	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1395	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  8138